arXiv:1602.06308v2 [math.CA] 19 Apr 2016
APPROXIMATION BY (p, q)-BASKAKOV-BETA OPERATORS
Neha Malik and Vijay Gupta Department of Mathematics, Netaji Subhas Institute of Technology Sector 3 Dwarka, New Delhi-110078, India neha.malik
[email protected] [email protected]
Abstract. In the present paper, we consider (p, q)-analogue of the Baskakov-Beta operators and using it, we estimate some direct results on approximation. Also, we represent the convergence of these operators graphically using MATLAB. Key Words. (p, q)-Beta function, (p, q)-Gamma function. AMS Subject Classification. 33B15, 41A25. 1. Introduction Approximation theory has been an engaging field of research with abstract approximation to the core (cf. [15]). Varied operators with their approximation properties, mainly the quantitative one, have been discussed and studied by many researchers. It has been seen that the generalizations of several well known operators to quantum-calculus (q-calculus) were introduced in the last three decades and their approximation behavior were also discussed (see [3], [10], [11] and [12]). Further generalization of quantum variant is the post-quantum calculus, denoted by (p, q)-calculus. Very recently, some researchers studied in this direction (see [4], [9] and [17]). Few basic definitions and notations mentioned below may be found in these papers and references therein. The (p, q)-numbers are given by [n]p,q := pn−1 + pn−2 q + pn−3 q 2 + · · · + pq n−2 + q n−1 pn −qn , if p 6= q 6= 1; p−q = n, if p = q = 1. The (p, q)-factorial is given by [n]p,q ! =
n Q r=1
[r]p,q , n > 1, [0]p,q ! = 1. The (p, q)-binomial
coefficient satisfies
n r
= p,q
[n]p,q ! , 0 6 r 6 n. [n − r]p,q ! [r]p,q !
Let n be a non-negative integer, the (p, q)-Gamma function is defined as Γp,q (n + 1) =
(p q)np,q = [n]p,q !, 0 < q < p, (p − q)n 1
where (p q)np,q = (p − q)(p2 − q 2 )(p3 − q 3 ) · · · (pn − q n ). The (p, q)-integral for 0 < q < p 6 1 (generalized Jackson integral) is defined as Za 0
i ∞ X qi aq f (x) dp,q x = (p − q)a f , x ∈ [0, a]. i+1 p pi+1 i=0
(1)
By simple computation, we get Za
xn dp,q x =
an+1 · [n + 1]p,q
0 i
q , i = 0, 1, . . . , geometrically Also, the integral (1) includes the nodes xi = xi (p, q) = pai+1 distributed in (0, +∞), not only in (0, a), as in the case p = 1 (standard Jackson’s qintegral). Moreover, one may observe that only a finite number of nodes in (1) are outside (0, a), i.e., those xi for which q i > pi+1 . Thus, the above definition of (p, q)-integral may be well utilized to define the (p, q)-extensions of well-known results. For m, n ∈ N, the (p, q)-Beta function of second kind considered in [2] is given by
Z∞ Bp,q (m, n) = 0
tm−1 dp,q t, (1 ⊕ pt)m+n p,q
where the (p, q)-power basis is given by (1 ⊕ pt)m+n = (1 + pt)(p + pqt)(p2 + pq 2 t) · · · (pm+n−1 + pq m+n−1 t). p,q Using the (p, q)-integration by parts: Zb
Zb f (px) Dp,q g(x) dp,q x = f (b) g(b) − f (a) g(a) −
a
g(qx) Dp,q f (x) dp,q x, a
it was shown in [2] that the following relation is satisfied by the (p, q)-analogues of Beta and Gamma functions: Bp,q (m, n) =
q Γp,q (m) Γp,q (n) (pm+1 q m−1 )m/2 Γp,q (m + n)
·
As a special case, if p = q = 1, B(m, n) = Γ(m) Γ(n)/Γ(m + n). It may be observed that in (p, q)-setting, order is important, which is the reason why (p, q)-variant of Beta function does not satisfy commutativity property, i.e., Bp,q (m, n) 6= Bp,q (n, m). For n ∈ N, x ∈ [0, ∞) and 0 < q < p 6 1, the (p, q)-analogue of Baskakov operators can be defined as n−1 ∞ X p [k]p,q p,q Bn,p,q (f, x) = bn,k (x)f , q k−1 [n]p,q k=0
where (p, q)-Baskakov basis function is given by xk n+k−1 p,q bn,k (x) = pk+n(n−1)/2 q k(k−1)/2 · k (1 ⊕ x)n+k p,q p,q Gupta [9] considered this form of (p, q)-Baskakov operators while studying its Kantorovich variant. This form was also considered by T. Acar et. al. in [5]. Remark 1. It has been observed in [9] that the (p, q)-Baskakov operators satisfy the following recurrence relation: p,q p,q p,q [n]p,q Tn,m+1 (qx) = q pn−1 x (1 + px) Dp,q [Tn,m (x)] + [n]p,q q x Tn,m (qx), ∞ m P pn−1 [k]p,q p,q (x) := Bn,p,q (em , x) = bp,q . where Tn,m n,k (x) q k−1 [n]p,q k=0
Then, we have Bn,p,q (e0 , x) = 1, Bn,p,q (e1 , x) = x, Bn,p,q (e2 , x) =
[n + 1]p,q x2 + pn−1 q x , q [n]p,q
where ei (t) = ti , i = 0, 1, 2. In case p = 1, we get the q-Baskakov operators [1], [11]. If p = q = 1, then these operators reduce to the well known Baskakov operators. 2. Construction of Operators and Moments In the year 1985, Sahai-Prasad [16] introduced the Durrmeyer variant of the well known Baskakov operators. However, there were some technical problems in the main estimates of [16], which were later improved by Sinha et. al. [19]. In this continuation, in 1994, Gupta proposed yet another Durrmeyer type generalization of Baskakov operators by taking the weights of Beta basis function. The operators discussed in [8] provide better approximation in simultaneous approximation than the usual Baskakov-Durrmeyer operators, studied in [19]. This motivated us to study further in this direction and here, we propose the (p, q)-variant of Baskakov-Beta operators. For n ∈ N, x ∈ [0, ∞) and 0 < q < p 6 1, the (p, q)-Baskakov-Beta operators are defined by: Z∞ ∞ X 1 tk p,q p,q Dn (f, x) = bn,k (x) f (q 2 pn+k t) dp,q t, (2) n+k+1 Bp,q (k + 1, n) (1 ⊕ pt)p,q k=0 0 n+k−1 xk where bp,q pk+n(n−1)/2 q k(k−1)/2 (1⊕x) n+k · n,k (x) = p,q k p,q In the present article, we estimate the moments of these operators by using (p, q)-Beta functions and establish some direct results in terms of modulus of continuity of first and second order using K-functional. Finally, we provide weighted approximation estimate alongwith the rate of convergence. Lemma 1. The following equalities hold:
(1) Dnp,q (1, x) = 1; x+pn−2 q , for n > 1; (2) Dnp,q (t, x) = [n]p,q[n−1] p,q (3) Dnp,q (t2 , x) =
n x2 [n]p,q [n]p,q + pq q [n−1]p,q [n−2]p,q
n−3
2
n−2
n−1 }
q +2 p q+p + x [n]p,q {p q [n−1]p,q [n−2]p,q
2n−5
p q [2]p,q + [n−1] , for n > 2. p,q [n−2]p,q
Proof. By Remark 1, we have
Dnp,q (1, x)
=
∞ X
bp,q n,k (x)
k=0
=
∞ X
=
Z∞ 0
bp,q n,k (x)
k=0 ∞ X
1 Bp,q (k + 1, n)
tk dp,q t (1 ⊕ pt)n+k+1 p,q
1 Bp,q (k + 1, n) Bp,q (k + 1, n)
bp,q n,k (x)
k=0
= Bn,p,q (1, x) = 1.
Next, using [k + 1]p,q = q k + p[k]p,q , we have
Dnp,q (t, x)
=
∞ X
bp,q n,k (x)
k=0
=
∞ X
=
k=0
Z∞ 0
bp,q n,k (x)
k=0 ∞ X
1 Bp,q (k + 1, n)
bp,q n,k (x)
tk+1 q 2 pn+k dp,q t (1 ⊕ pt)n+k+1 p,q
1 q 2 pn+k Bp,q (k + 2, n − 1) Bp,q (k + 1, n) q [2−(k+1)k]/2
Γp,q (n + k + 1) −(k+1)(k+2)/2 p Γp,q (k
× q [2−(k+2)(k+1)]/2 p−(k+2)(k+3)/2 =
∞ X k=0
n−2 1−k bp,q q n,k (x) p
+ 1) Γp,q (n)
q 2 pn+k
Γp,q (k + 2) Γp,q (n − 1) Γp,q (n + k + 1)
[k + 1]p,q [n − 1]p,q
∞ X 1 = bp,q (x) pn−2 q 1−k (q k + p[k]p,q ) [n − 1]p,q k=0 n,k ∞ ∞ X pn−2 q X p,q [n]p,q pn−1 [k]p,q = bn,k (x) + bp,q (x) · [n − 1]p,q k=0 [n − 1]p,q k=0 n,k q k−1 [n]p,q
Using Remark 1, we have
[n]p,q pn−2 q Bn,p,q (1, x) + Bn,p,q (t, x) [n − 1]p,q [n − 1]p,q pn−2 q [n]p,q = ·1+ ·x [n − 1]p,q [n − 1]p,q [n]p,q x + pn−2 q = · [n − 1]p,q
Dnp,q (t, x) =
Further, using the identity [k + 2]p,q = q k+1 + p q k + p2 [k]p,q , we get
Dnp,q (t2 , x)
=
∞ X
bp,q n,k (x)
k=0
=
∞ X
=
k=0
Z∞ 0
bp,q n,k (x)
k=0 ∞ X
1 Bp,q (k + 1, n)
bp,q n,k (x)
tk+2 q 4 p2(n+k) dp,q t (1 ⊕ pt)n+k+1 p,q
1 q 4 p2(n+k) Bp,q (k + 3, n − 2) Bp,q (k + 1, n) q [2−(k+1)k]/2
Γp,q (n + k + 1) −(k+1)(k+2)/2 p Γp,q (k
× q [2−(k+3)(k+2)]/2 p−(k+3)(k+4)/2 =
∞ X k=0
2n−5 1−2k bp,q q n,k (x) p
q 4 p2(n+k)
Γp,q (k + 3) Γp,q (n − 2) Γp,q (n + k + 1)
[k + 2]p,q [k + 1]p,q [n − 1]p,q [n − 2]p,q
∞ X 1 = bp,q (x) p2n−5 q 1−2k [n − 1]p,q [n − 2]p,q k=0 n,k × q k + p [k]p,q ∞ p2n−5 q 2 + p2n−4 q X p,q b (x) = [n − 1]p,q [n − 2]p,q k=0 n,k
+
+ 1) Γp,q (n)
q k+1 + p q k + p2 [k]p,q
∞ X pn−3 q [n]p,q pn−1 [k]p,q bp,q (x) [n − 1]p,q [n − 2]p,q k=0 n,k q k−1 [n]p,q
∞ X 2 pn−2 [n]p,q pn−1 [k]p,q + bp,q (x) [n − 1]p,q [n − 2]p,q k=0 n,k q k−1 [n]p,q ∞ X [n]2p,q p2n−2 [k]2p,q + bp,q (x) · q [n − 1]p,q [n − 2]p,q k=0 n,k q 2k−2 [n]2p,q
Again, using Remark 1 pn−3 q [n]p,q p2n−5 q 2 + p2n−4 q Bn,p,q (1, x) + Bn,p,q (t, x) [n − 1]p,q [n − 2]p,q [n − 1]p,q [n − 2]p,q [n]2p,q 2 pn−2 [n]p,q Bn,p,q (t, x) + Bn,p,q (t2 , x) + [n − 1]p,q [n − 2]p,q q [n − 1]p,q [n − 2]p,q p2n−5 q 2 + p2n−4 q x [n]p,q {pn−3 q + 2 pn−2 } = + [n − 1]p,q [n − 2]p,q [n − 1]p,q [n − 2]p,q n−1 [n]2p,q p x px 2 + x + 1+ [n]p,q q q [n − 1]p,q [n − 2]p,q n x2 [n]p,q [n]p,q + pq x [n]p,q {pn−3 q 2 + 2 pn−2 q + pn−1 } = + q [n − 1]p,q [n − 2]p,q q [n − 1]p,q [n − 2]p,q 2n−5 p q [2]p,q + · [n − 1]p,q [n − 2]p,q
Dnp,q (t2 , x) =
Remark 2. Let n > 2 and x ∈ [0, ∞), then for 0 < q < p 6 1, we have the central moments as follows: p,q µp,q n,1 (x) := Dn ((t − x), x)
x ([n]p,q − [n − 1]p,q ) + pn−2 q [n − 1]p,q
= and p,q 2 µp,q n,2 (x) := Dn ((t − x) , x) n x2 [n]p,q [n]p,q + =
+
x{[n]p,q (p
n−3
pn q
+ q [n − 1]p,q [n − 2]p,q − 2 q [n]p,q [n − 2]p,q
o
q [n − 1]p,q [n − 2]p,q [2]p,q p2n−5 q q + 2p q + pn−1 ) − 2 pn−2 q 2 [n − 2]p,q } + · q [n − 1]p,q [n − 2]p,q [n − 1]p,q [n − 2]p,q 2
n−2
3. Direct Estimations In this section, we prove direct results using two different approaches, i.e., K-functional and Steklov mean. We also represent the convergence of the (p, q)-Baskakov-Beta operators using the software MATLAB. We denote the norm ||f || = sup |f (x)| on CB [0, ∞), the class of real valued continuous x∈[0,∞)
bounded functions. For f ∈ CB [0, ∞) and δ > 0, the m-th order modulus of continuity is defined as ωm (f, δ) = sup sup |∆m h f (x)|, 06h6δ x∈[0,∞)
m−1 where ∆h is the forward difference and ∆m for m > 1. In case m = 1, we h = ∆h ∆h mean the usual modulus of continuity denoted by ω(f, δ). The Peetre’s K-functional is defined as 00 2 K2 (f, δ) = inf ||f − g|| + δ||g || : g ∈ C [0, ∞) , B 2 g∈CB [0,∞)
where CB2 [0, ∞) = {g ∈ CB [0, ∞) : g 0 , g 00 ∈ CB [0, ∞)}. Theorem 1. Let f ∈ CB [0, ∞) and 0 < q < p 6 1, then for every x ∈ [0, ∞) and n > 2, the following inequality holds: q 2 p,q p,q p,q p,q , |Dn (f, x) − f (x)| 6 ω f, |µn,1 (x)| + C ω2 f, µn,2 (x) + µn,1 (x)
where C is some positive constant. Proof. Consider the following operator: [n]p,q x + pn−2 q p,q p,q ˇ Dn (f, x) = Dn (f, x) − f + f (x), x ∈ [0, ∞) [n − 1]p,q
(3)
Let g ∈ CB2 [0, ∞) and x, t ∈ [0, ∞). By Taylor’s expansion, we have g(t) = g(x) + (t − x) g 0 (x) +
Zt
(t − u) g 00 (u) du,
x
ˇ np,q , we get Applying D
Zt
ˇ p,q (g, x) − g(x) = g 0 (x) D ˇ p,q ((t − x), x) + D ˇ p,q (t − u) g 00 (u) du, x . D n n n x
Hence, ˇ p,q (g, x) − g(x)| |D n
6
Zt p,q 00 Dn |t − u| |g (u)| du , x +
[n]p,q x+pn−2 q [n−1]p,q
Z
x
[n]p,q x + pn−2 q 00 − u g (u) du [n − 1]p,q
x [n]p,q x+pn−2 q [n−1]p,q
[n]p,q x + pn−2 q 00 6 − u g (u) du [n − 1]p,q x ( 2 ) x ([n]p,q − [n − 1]p,q ) + pn−2 q p,q 6 µn,2 (x) + kg 00 k. [n − 1]p,q Dnp,q ((t
− x) , x) kg k + 2
00
Z
Now, using operators (2), we have, by Lemma 1, ∞ X
1 |Dnp,q (f, x)| 6 bp,q n,k (x) Bp,q (k + 1, n) k=0
Z∞ 0
tk |f (q 2 pn+k t)| dp,q t. (1 ⊕ pt)n+k+1 p,q
Hence, by (3), |Dnp,q (f, x)| 6 3 kf k. Therefore |Dnp,q (f, x) − f (x)| n−2 [n] x + p q p,q p,q p,q ˇ ˇ − f (x) 6 |Dn (f − g, x) − (f − g)(x)| + |Dn (g, x) − g(x)| + f [n − 1]p,q ( ) 2 x ([n]p,q − [n − 1]p,q ) + pn−2 q 6 4 kf − gk + µp,q (x) + kg 00 k n,2 [n − 1]p,q |x ([n]p,q − [n − 1]p,q ) + pn−2 q| . + ω f, [n − 1]p,q Lastly, √ taking infimum over all g ∈ CB2 [0, ∞), and using the inequality K2 (f, δ) 6 C ω2 (f, δ), δ > 0 due to [6], we get the desired assertion. Example 1. We show comparisons and some illustrative graphs for the convergence of (p, q)-analogue of Baskakov-Beta operators Dnp,q (f, x) for different values of the parameters p and q, such that 0 < q < p 6 1. For x ∈ [0, ∞), p = 0.9 and q = 0.8, the convergence of the operators Dnp,q (f, x) to the function f , where f (x) = 18x2 − 12x + 2015, for different values of n is illustrated using MATLAB.
Figure 1. Dn0.9,0.8 (f, x) for x ∈ [0, ∞), when f (x) = 18x2 − 12x + 2015.
Example 2. For x ∈ [0, ∞), p = 0.9 and q = 0.75, the convergence of the operators Dnp,q (f, x) to the function f , where f (x) = 25x2 − 2x + 7, for different values of n is illustrated using MATLAB.
Figure 2. Dn0.9,0.75 (f, x) for x ∈ [0, ∞), when f (x) = 25x2 − 2x + 7.
Let Bσ [0, ∞) be the space of all real valued functions on [0, ∞) satisfying the condition |f (x)| 6 Cf σ(x), where Cf > 0 and σ(x) is a weight function. Let Cσ [0, ∞) be the (x)| space of all continuous functions in Bσ [0, ∞) with the norm kf kσ = sup |fσ(x) and x∈[0,∞) o n (x)| < ∞ . We consider σ(x) = (1+x2 ) in the following Cσ0 [0, ∞) = f ∈ Cσ [0, ∞) : lim |fσ(x) x→∞
two results. Also, we denote the modulus of continuity on f on the closed interval [0, κ], κ > 0 by ωκ (f, δ) = sup |t − x| 6 δ sup |f (t) − f (x)|. x,t∈[0,κ]
It is easy to see that for f ∈ Cσ [0, ∞), the modulus of continuity ωκ (f, δ) tends to zero. Following is a theorem on the rate of convergence for the operators Dnp,q (f, x). Theorem 2. Let f ∈ Cσ [0, ∞), 0 < q < p 6 1 and ωκ+1 (f, δ) be its modulus of continuity on the finite interval [0, κ + 1] ⊂ [0, ∞), where κ > 0. Then, for every n > 2, q p,q ||Dnp,q (f ) − f ||C[0,κ] 6 L µp,q (x) + ω + 2 ω f, L µ (x) , κ+1 n,2 n,2 where L = 6 Cf (1 + κ2 )(1 + κ + κ2 ). Proof. For x ∈ [0, κ] and t > κ + 1, since t − x > 1, therefore we have |f (t) − f (x)| 6 Cf (2 + x2 + t2 ) 6 (2 + 3x2 + 2(t − x)2 ) 6 6 Cf (1 + κ2 )(t − x)2 .
(4)
And for x ∈ [0, κ] and t 6 κ + 1, we have |t − x| ωκ+1 (f, δ), |f (t) − f (x)| 6 ωκ+1 (f, |t − x|) 6 1 + δ with δ > 0. Using above two equations (4) and (5), |t − x| 2 2 |f (t) − f (x)| 6 6 Cf (1 + κ )(t − x) + 1 + ωκ+1 (f, δ), δ
(5)
for t > 0. Now, |Dnp,q (f, x) − f (x)| 6 Dnp,q (|f (t) − f (x)|, x) 2
6 6 Cf (1 + κ
)Dnp,q ((t
2
− x) , x) + ωκ+1 (f, δ)
1 1 + Dnp,q ((t − x)2 , x) δ
Using Remark 2 and Schwarz’s inequality, for every 0 < q < p 6 1 and x ∈ [0, κ], we have
|Dnp,q (f, x)
Taking δ =
1 q p,q − f (x)| 6 6 Cf (1 + κ ) + ωκ+1 (f, δ) 1 + µn,2 (x) δ 1 q p,q p,q 6 L µn,2 (x) + ωκ+1 (f, δ) 1 + µn,2 (x) . δ 2
µp,q n,2 (x)
q L µp,q n,2 (x), the conclusion holds.
Following is a direct estimate in weighted approximation. Theorem 3. Let p = pn , q = qn satisfying 0 < qn < pn 6 1 and pn → 1, qn → 1, pnn → a, qnn → b as n → ∞. Then, for f ∈ Cσ0 [0, ∞), we have lim kDnp,q (f ) − f kσ = 0.
n→∞
Proof. Due to the well-known Bohman-Korovkin theorem in [7], it is sufficient to verify the following equation holds: lim kDnp,q tk , x − xk kσ = 0, for k = 0, 1, 2.
n→∞
By Lemma 1, the result immediately follows for k = 0. Again, by Lemma 1, we have:
kDnp,q (t, x)
− xkσ
[n]p,q x + pn−2 q 1 = sup − x [n − 1]p,q (1 + x2 ) x∈[0,∞) x ([n]p,q − [n − 1]p,q ) + pn−2 q 1 = sup [n − 1]p,q (1 + x2 ) x∈[0,∞)
1/2
and kDnp,q (t2 , x) − x2 kσ =
n x2 [n] [n]p,q + p,q sup
pn q
x∈[0,∞)
+ q [n − 1]p,q [n − 2]p,q − 2 q [n]p,q [n − 2]p,q q [n − 1]p,q [n − 2]p,q
x {[n]p,q (pn−3 q 2 + 2 pn−2 q + pn−1 ) − 2 pn−2 q 2 [n − 2]p,q } q [n − 1]p,q [n − 2]p,q 2n−5 1 [2]p,q p q 2 + −x [n − 1]p,q [n − 2]p,q (1 + x2 ) n o pn x2 [n] − 2 q [n]p,q [n − 2]p,q p,q [n]p,q + q = sup q [n − 1]p,q [n − 2]p,q x∈[0,∞) +
x {[n]p,q (pn−3 q 2 + 2 pn−2 q + pn−1 ) − 2 pn−2 q 2 [n − 2]p,q } q [n − 1]p,q [n − 2]p,q 2n−5 [2]p,q p q 1 + , [n − 1]p,q [n − 2]p,q (1 + x2 ) +
which shows that the result holds for k = 1 and k = 2 as well. This completes the proof of the theorem.
Remark 3. In order to obtain convergence estimates in the theorems discussed in the present article, we consider 0 < qn < pn 6 1, p = pn and q = qn , such that lim pn = 1 = n→∞
lim qn and lim pnn = a, lim qnn = b. Thus, we have lim
n→∞
n→∞
1
n→∞ [n]pn ,qn
n→∞
= 0.
Remark 4. The Srivastava-Gupta operators came into existence in the year 2003 (cf. [18]). Several further generalizations of these well-known operators were discussed (cf. [13], [14], [20] and [21]). The (p, q)-variant of these operators cannot be defined for general form. Although, one can define the (p, q)-extensions for special cases of these operators separately. This leads to an open problem for our readers. Acknowledgement. The authors are grateful to Prof. G. V. Milovanovi´ c for valuable discussions concerning the (p, q)-integral and their properties. References [1] A. Aral and V. Gupta, Generalized q-Baskakov operators, Math. Slovaca 61, 4 (2011), 619–634. [2] A. Aral and V. Gupta, (p, q)-type Beta functions of second kind, communicated. [3] A. Aral, V. Gupta and R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, 2013. [4] T. Acar, (p, q)-Generalization of Sz´ asz-Mirakyan Operators, arXiv preprint arXiv:1505.06839. [5] T. Acar, A. Aral and S. A. Mohiuddine, On Kantorovich modification of (p, q)-Baskakov operators, J. Inequal. Appl. Article ID 98, (2016), 14 pages. [6] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, Berlin, 1993. [7] A. D. Gadzhiev, Theorems of the type of P. P. Korovkin type theorems, Mat. Zametki 20, 5 (1976), 781–786; English Translation, Math Notes 20, (5-6) (1976), 996–998. [8] V. Gupta, A note on modified Baskakov type operators, Approx. Theory Appl. 10, 3 (1994), 74–78. [9] V. Gupta, (p, q)-Baskakov-Kantorovich operators, Appl. Math. Inf. Sci. 10, 4 (2016).
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[10] V. Gupta, Some approximation properties on q-Durrmeyer operators, Appl. Math. Comput. 197, 1 (2008), 172–178. [11] V. Gupta and Z. Finta, On certain q-Durrmeyer operators, Appl. Math. Comput. 209, (2009), 415–420. [12] V. Gupta and R. P. Agarwal, Convergence Estimates in Approximation Theory, Springer, 2014. [13] N. Ispir and I. Yuksel, On the B´ ezier variant of Srivastava-Gupta operators, Appl. Math. E-Notes 5, (2005), 129–137. [14] N. Malik, Some approximation properties for generalized Srivastava-Gupta operators, Appl. Math. Comput. 269, (2015), 747–758. [15] M. S. Moslehian and H. Najafi, Around operator monotone functions, Integr. Equ. Oper. Theory 71, (2011), 575–582. [16] A. Sahai and G. Prasad, On simultaneous approximation by modified Lupa¸s operators, J. Approx. Theory 45, (1985), 122–128. [17] V. Sahai and S. Yadav, Representations of two parameter quantum algebras and p, q-special functions, J. Math. Anal. Appl. 335, (2007), 268–279. [18] H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37 (12),(2003), 1307–1315. [19] R. P. Sinha, P. N. Agrawal and V. Gupta, On simultaneous approximation by modified Baskakov operators, Bull. Soc. Begl. Ser. B. 43, (1991), 217–231. [20] D. K. Verma and P. N. Agrawal, Convergence in simultaneous approximation for SrivastavaGupta operators, Mathematical Sciences Vol. 6, Article 22, (2012). [21] R. Yadav, Approximation by modified Srivastava-Gupta operators, Appl. Math. Comput. 226, (2014), 61–66.
